F.A.Q:

Q. How did you do the 3D animations?
A. I programmed them in Python, using the manim library: https://www.manim.community

Q. How can I play around with these types of graphs myself?
A. Any software that can graph 3D parametric curves will do, but you do have to define the real and imaginary parts of the parametric function yourself. Here's an interactive example for x^a using GeoGebra 3D: https://www.geogebra.org/m/wtqwhswu

Q. Isn't x^2.4 just the 5th-degree root of x^12? (and thus just a negative number for negative x?)
A. That is one possible way to interpret it, yes. There are other just-as-valid interpretations. This is what I wanted you to think about with the suggested exercise at 13:50. When the exponent a in x^a can be written as a fraction with an odd base (which is the case for 2.4 = 12/5) then one of the many values of x^a will be a real number, even for negative x. Which one of the multiple values to pick from can be subject to debate, and you can see that in the showcase starting at 00:23 only one of the three graphing tools that we tested (Desmos) chooses this interpretation of always assigning a real value to x^a when possible.

Q. So you are defining x^a by using e^something-else? Isn't that a circular definition?
A. No. The function e^x or exp(x) is not understood to be an exponentiation, but rather a function defined by its series expansion (this expansion can be seen at 07:07).

Q. Where do all of these definitions come from anyway?
A. That's a great question, hopefully for another video.

Bro dropped a 3B1B caliber video, it blew up, and then hasn't uploaded another one since? 6.7k subs? 28k likes? Crazy channel honestly. Hope he comes up in my feed soon with something new.

Complex topics don't have to be dumbed down, all it takes is a good visualisation and it can be taught to anyone.

My jaw hit the floor when x^a started rotating around the real axis and suddenly I understood exactly why even a curves up and odd a curves down. Fantastic video, thanks for sharing these concepts!

Bro just posted this gem two years ago and never posted again

I love how math is always self consistent. Literally spins around in the complex plane to get to where it needs to go

This is HANDS DOWN the best of the 10 or so #SoME2 videos I've watched so far. 1/ It centers around a "simple" (middle/high school) topic: graphing some of the most basic functions. 2/ It reveals beautiful hidden aspects of the topic that are likely surprising to many people, even those with a higher degree in STEM (usually because the visualization of complex functions is HIGHLY underexplored for the sake of rushing to more "interesting" topics in complex analysis). 3/ It encourages the audience to experiment, think about different approaches, and explore exercises. 4/ It is well produced (great visualizations, great audio quality, great script).
For my taste, you could have spent a little more time on the general formula for any x in C* and the more technical parts involving algebraic manipulations and the different branches of the logarithm (see how carefully 3b1b does it ;-) ), but in general I think that shouldn't bar you from being one of the top contenders for the win.
Keep up the good work!

Bro randomly chose to post a math video after posting Mario 64 content and chose to disappear

Looked like that's some high level mathematics easily explained to a 16 yr dumb like me . Thank you sir for such a interesting session

The fun thing is how complex numbers have this tendency to create rotations.

Essentially when you have an irrational exponent you are taking a root.

Bud, this is incredible. Why did this just get recommended to me? I suddenly understand things I wish I had known decades ago!

But why nobody tries to continue the idea of 13:11 graph and draw y=x^x at all k? It will look like a surface (like Riemann surface), and probably is the most correct visualization

This answers a problem my buddy and I ran into in our high school Calc I class (effectively (-x)^a; the best we could do was to determine this was a way to determine “whether a fraction is odd or even”).

I’ve not done much higher level math since then, but randomly revisited it about a year ago and realized there was probably something in the imaginary plane that this could reveal, but lacked the familiarity with the tools to take it much further than that.

All that is to say: thanks for the video. It answered some long standing questions I’ve had about this type of question.

When I went for my undergrad, I had to take analysis, which jumped straight from R -> R into C -> C without any visual assistance for how to interpret the latter.  I think getting some exposure to R -> C (and following that up with R+ki -> C) would be a really good way to get a handle on why n-dimensional space needs to give way to transformations as the best visualization, as you move towards C -> C.  The fact that R -> C can be informative in both frames without one or the other being obviously better makes it a really useful example.

The difference between dumbing down for the viewer and raising the viewer up to the desired level is such a shockingly pleasant one to experience. Brilliantly satisfying video, even a comforting one in its beauty.

A 15 minute video explained what 15 years at school could not. Good job !!

12:48 It reminds me of the "All models are wrong, some are useful" quote. The more accurate models are good, but depending on what you need some details are not as important.

5:26 my phone charger when I plug it in upside-down from how I did last time

To fully describe the emotions I experienced, I need to combine the sensation of drinking water after a long thirst in the desert with an orgasm. Finally...  THE ACCURACY

I absolutely LOVED this video. It's no exaggeration to say that the beauty of those curves on in R X C (especially when "animated") nearly moves me to tears. Thank you very much.